New Results on Path Approximation

Abstract

In this paper we give bounds on the complexity of some algorithms for approximating 2-D and 3-D polygonal paths with the infinite beam measure of error. While the time/space complexities of the algorithms known for other error measures are well understood, path approximation with infinite beam measure seems to be harder due to the complexity of some geometric structures that arise in the known approaches. Our results answer some open problems left in previous work. We also present a more careful analysis of the time complexity of the general iterative algorithm for infinite beam measure and show that it could perform much better in practice than in the worst case. We then propose a new approach for path approximation that consists of a breadth first traversal of the path approximation graph, without constructing the graph. This approach can be applied to any error criterion in any constant dimension. The running time of the new algorithm depends on the size of the output: if the optimal approximating path has m vertices, the algorithm performs F(m) iterations rather than n − 1 in the previous approaches, where F(m) ≤ n − 1 is the number of vertices of the path approximation graph that can be reached with at most m − 2 edges. This is the first output sensitive path approximation algorithm.

Optimal simplification of polygonal chain for rendering

“…This method achieves efficiency but fails to build an approximation with a minimum number of vertices. Other recent approaches, a breadth-first traversal of the graph [7], a query-based technique [8] (for the infinite beam criterion) and an error measure based on the Fréchet distance [2] achieve near-linear time performance but only under certain assumptions.…”

Review and classification of trajectory summarisation algorithms: From compression to segmentation

“…Coresets, 34 AACAT, 30 SimpleTrack, 31 SGTCR-CS 27 Probabilistic IMM, [35][36][37] APSOS, 38 SAS, 32 SAOTS, 33 SGTCR-CS 27 Graph Distance Bellman, 39,40 DOTS, 41 DOTS-CASCADE, 41 Iri-Imai, 42,43 MRPA, 44 Daescu, 45,46 OGPC and OSPC, 47 MMTC-offline, 48 MMTC-online, 48 SPPA, 49 GRTSOpt, 50 Latecki, 51 Trajic, 52 Representativeness, 53 KAA and StreamKAA, 54 OLTS and OPTTS, 55 DOTS*, 56 OSC and OSTC, 28 CLEAN 57 Angle VTracer, 58 DPTS + , 59 Latecki, 51 61 GRPPA, 62 TSHL, 63 AMS, 16 CFF, 64 BOPW and NOPW, 4 OHTA, OnlineOHTA and SATA, 65 CDR, CDRm, GRTSOpt and GRTSSec, 50 TraClus, 66 OPERB and A-OPERB, 67 BQS, 68 ABQS, FBQS and PBQS, 69 LO-OPW-TR, 70 OPW-TR, 3 SMoT, 71 Pan, 72 Patroumpas,…”

Generating parametric models of tubes from laser scans

“…The algorithm is based on a general framework used in several polygon simplification algorithms [12,25]. The main idea is to construct a shortcut graph G = (V, E) over the vertices of the input curve, with an edge e i j ∈ E if and only if the chain [p i , p i+1 , .…”

“…The graph does not have to be constructed explicitly, therefore our algorithm only requires O(n) space, see [25].…”

“…We used the algorithm from [25], which computes a minimum segment approximation using a subset of the input vertices with distance less or equal to in time O(n 3 ), using space O(n). We observed a real-world running time in the order of magnitude of about O(n 2 ).…”